Class 6 — Mathematics (NCERT)

Playing with Numbers

Class 6

✓By the end of this lesson students will be able to identify factors and multiples of given numbers.
✓By the end of this lesson students will be able to distinguish between prime and composite numbers.
✓By the end of this lesson students will be able to perform prime factorisation of numbers.
✓By the end of this lesson students will be able to find the Highest Common Factor (HCF) of two or more numbers.
✓By the end of this lesson students will be able to find the Lowest Common Multiple (LCM) of two or more numbers.

Key concepts

Factors

A factor of a number is an exact divisor of that number. When a number is divided by its factor, the remainder is zero. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

Properties of Factors

1. 1 is a factor of every number.\n2. Every number is a factor of itself.\n3. Every factor of a number is an exact divisor of that number.\n4. Every factor is less than or equal to the given number.\n5. The number of factors of a given number is finite.

Multiples

A multiple of a number is a number obtained by multiplying it by some natural number (1, 2, 3, ...). For example, the multiples of 5 are 5, 10, 15, 20, ...

Properties of Multiples

1. Every number is a multiple of itself.\n2. Every multiple of a number is greater than or equal to that number.\n3. The number of multiples of a given number is infinite.

Prime Numbers

A prime number is a natural number greater than 1 that has exactly two factors: 1 and the number itself. Examples: 2, 3, 5, 7, 11, etc.

Composite Numbers

A composite number is a natural number greater than 1 that has more than two factors. Examples: 4, 6, 8, 9, 10, etc. Note: The number 1 is neither prime nor composite.

Twin Primes

Two prime numbers are called twin primes if there is only one composite number between them. In other words, they are prime numbers that differ by 2. Examples: (3, 5), (5, 7), (11, 13).

Co-prime Numbers

Two numbers are said to be co-prime (or relatively prime) if their only common factor is 1. Note that co-prime numbers need not be prime numbers themselves. Examples: (4, 9), (8, 15).

Perfect Numbers

A number is called a perfect number if the sum of all its factors (excluding the number itself) is equal to the number itself. Examples: 6 (factors are 1, 2, 3, 6; 1+2+3 = 6), 28 (factors are 1, 2, 4, 7, 14, 28; 1+2+4+7+14 = 28).

Prime Factorisation

Prime factorisation is the process of expressing a composite number as a product of its prime factors. This can be done using a factor tree method or a division method.

Highest Common Factor (HCF)

The Highest Common Factor (HCF) of two or more given numbers is the largest of their common factors. It is also known as the Greatest Common Divisor (GCD).

Lowest Common Multiple (LCM)

The Lowest Common Multiple (LCM) of two or more given numbers is the smallest non-zero common multiple of those numbers.

Key facts to remember

11 is a factor of every number and is neither prime nor composite.
2Every number is a multiple of itself.
3The smallest prime number is 2, and it is the only even prime number.
4The smallest composite number is 4.
5Factors are finite, while multiples are infinite.
6The HCF of two or more numbers is always less than or equal to the smallest of the given numbers.
7The LCM of two or more numbers is always greater than or equal to the largest of the given numbers.
8For any two numbers, Product of the two numbers = HCF x LCM.

Worked examples

Example 1

Find all factors of 36. Then, classify 36 as a prime or composite number. Also, list the first five multiples of 36.

ITo find factors of 36, we look for pairs of numbers whose product is 36:

II1 x 36 = 36

III2 x 18 = 36

IV3 x 12 = 36

V4 x 9 = 36

VI6 x 6 = 36

VIISo, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

VIIISince 36 has more than two factors (it has 9 factors), it is a composite number.

9To find the first five multiples of 36, we multiply 36 by 1, 2, 3, 4, and 5:

1036 x 1 = 36

1136 x 2 = 72

1236 x 3 = 108

1336 x 4 = 144

1436 x 5 = 180

Answer

Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. 36 is a composite number. The first five multiples of 36 are 36, 72, 108, 144, 180.

Example 2

Find the HCF of 48 and 72 using the prime factorisation method.

IStep 1: Perform prime factorisation for each number.

IIFor 48:

III48 = 2 x 24

IV = 2 x 2 x 12

V = 2 x 2 x 2 x 6

VI = 2 x 2 x 2 x 2 x 3

VIISo, 48 = 2⁴ x 3¹

VIIIFor 72:

972 = 2 x 36

10 = 2 x 2 x 18

11 = 2 x 2 x 2 x 9

12 = 2 x 2 x 2 x 3 x 3

13So, 72 = 2³ x 3²

14Step 2: Identify the common prime factors and their lowest powers.

15Common prime factors are 2 and 3.

16Lowest power of 2 is 2³ (from 72).

17Lowest power of 3 is 3¹ (from 48).

18Step 3: Multiply these common prime factors with their lowest powers.

19HCF = 2³ x 3¹

20 = 8 x 3

21 = 24

Answer

The HCF of 48 and 72 is 24.

Remember to take the lowest power of each common prime factor when finding HCF.

Example 3

Find the LCM of 12, 18, and 24 using the prime factorisation method.

IStep 1: Perform prime factorisation for each number.

IIFor 12:

III12 = 2 x 6

IV = 2 x 2 x 3

VSo, 12 = 2² x 3¹

VIFor 18:

VII18 = 2 x 9

VIII = 2 x 3 x 3

9So, 18 = 2¹ x 3²

10For 24:

1124 = 2 x 12

12 = 2 x 2 x 6

13 = 2 x 2 x 2 x 3

14So, 24 = 2³ x 3¹

15Step 2: Identify all prime factors involved and their highest powers.

16The prime factors involved are 2 and 3.

17Highest power of 2 is 2³ (from 24).

18Highest power of 3 is 3² (from 18).

19Step 3: Multiply these prime factors with their highest powers.

20LCM = 2³ x 3²

21 = 8 x 9

22 = 72

Answer

The LCM of 12, 18, and 24 is 72.

Remember to take the highest power of all prime factors (common or not) when finding LCM.

Common mistakes

✗Confusing factors with multiples (e.g., saying 10 is a factor of 5 instead of a multiple of 5).
✗Incorrectly identifying 1 as a prime or composite number.
✗Errors in prime factorisation, leading to incorrect HCF or LCM.
✗Forgetting to include all prime factors (with their highest powers) when calculating LCM using prime factorisation.
✗Forgetting to take only common prime factors (with their lowest powers) when calculating HCF using prime factorisation.

Exam tips

★Always show your working steps clearly, especially for prime factorisation, HCF, and LCM calculations.
★For HCF and LCM, if the numbers are small, listing factors/multiples can be a quick check, but for larger numbers, prime factorisation is more reliable.
★Double-check your prime factorisation to ensure all factors are indeed prime numbers.
★Practice word problems involving HCF and LCM to understand when to apply each concept (HCF for 'greatest', 'maximum', 'largest group'; LCM for 'least', 'minimum', 'first time they meet again').

Ready to practise?

Try a problem on this topic

Snap a photo or type a question — get step-by-step working instantly.

Solve a problem →← Maths curriculum